Applications of group theory in quantum mechanics

www.pdfgrip.com Copyright Copyright © 1969 by M I Petrashen and E D Trifonov All rights reserved Bibliographical Note This Dover edition, first published in 2009, is an unabridged republication of the work first published in English in 1969 by The M.I.T Press, Cambridge, Massachusetts It was originally published in Moscow under the title Primeneniye Teorii Grupp v Kvantovoi Mekhanike Library of Congress Cataloging-in-Publication Data Petrashen, M I (Mariia Ivanovna) [Primenenie teorii grupp v kvantovoi mekhanike English] Applications of group theory in quantum mechanics / M I Petrashen and E.D Trifonov — Dover ed p cm Originally published: Cambridge, Mass : M.I.T., 1969 Includes bibliographical references and index 9780486172729 Group theory Quantum theory I Trifonov, E D (Evgenii Dmitrievich), joint author II Title www.pdfgrip.com QA174.2.P4813 2008 530.1201’5122 — dc22 2008044542 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y 11501 www.pdfgrip.com Table of Contents Title Page Copyright Page Foreword Chapter - Introduction Chapter - Abstract Groups Chapter - Representations of Point Groups Chapter - Composition of Representations and the Direct Products of Groups Chapter - Wigner’s Theorem Chapter - Point Groups Chapter - Decomposition of a Reducible Representation into an Irreducible Representation Chapter - Space Groups and Their Irreducible Representations Chapter - Classification of the Vibrational and Electronic States of a Crystal Chapter 10 - Continuous Groups Chapter 11 - Irreducible Representations of the Three—Dimensional Rotation Group Chapter 12 - The Properties of Irreducible Representations of the Rotation Group Chapter 13 - Some Applications of the Theory of Representation of the Rotation Group in Quantum Mechanics Chapter 14 - Additional Degeneracy in a Spherically Symmetric Field www.pdfgrip.com Chapter 15 - Permutation Groups Chapter 16 - Symmetrized Powers of Representations Chapter 17 - Symmetry Properties of Multi-Electron Wave Functions Chapter 18 - Symmetry Properties of Wave Functions for a System of Identical Particles with Arbitrary Spins Chapter 19 - Classification of the States of a Multi-Electron Atom Chapter 20 - Applications of Group Theory To Problems Connected With the Perturbation Theory Chapter 21 - Selection Rules Chapter 22 - The Lorentz Group and its Irreducible Representations Chapter 23 - The Dirac Equation Appendix to Chapter Bibliography Index www.pdfgrip.com Foreword This monograph is based on a course of lectures on the applications of group theory to problems in quantum mechanics, given by the authors to undergraduates at the Physics Department of Leningrad University Following a period of scepticism about the value of group theory as a means of investigating physical systems, this mathematical theory eventually won a very general acceptance by physicists The group-theory formalism is now widely used in various branches of quantum physics, including the theory of the atom, the theory of the solid state, quantum chemistry, and so on Recent achievements in the theory of elementary particles, which are intimately connected with the application of group theory, have intensified general interest in the possibility of using group-theoretical methods in physics, and have shown once again the importance and eminent suitability of such methods in quantum theory A relatively large number of textbooks and monographs on applications of group theory in physics is already available A bibliography is given at the end of the book The range of applications of the methods of group theory to physics is continually expanding, and it is hardly possible at the present time to produce a monograph which would cover all these applications The best course to adopt, therefore, is to include the relevant applications in monographs or www.pdfgrip.com textbooks devoted to special topics in physics This is done, for example, in the well-known course on theoretical physics by Landau and Lifshits It is likely that this tendency will continue in the future At the same time, a theoretical physicist should have a general knowledge of the leading ideas and methods of group theory as used in physics Our aim in this course was to satisfy this need Moreover, we thought it would be useful to include in the book a number of problems which have not been discussed in existing monographs, or treated in sufficient detail We refer, above all, to studies of the symmetry properties of the Schroedinger wave function, to the explanation of ‘additional’ degeneracy in the Coulomb field, and to certain problems in solid-state physics In our course, we have restricted our attention to applications of group theory to quantum mechanics It follows that the book can be regarded as the first part of a broader course, the second part of which should be devoted to applications of group-theoretical methods to quantum field theory We conclude our book with an account of related problems concerned with the conditions for relativistic invariance in quantum theory We are grateful to M N Adamov, who read this monograph in manuscript and made a number of valuable suggestions, and to A G Zhilich and I B Levinson, who reviewed individual chapters In the preparation of the manuscript for press we made use of the kind assistance of A A Kiselev, B Ya Frezinskii, R A Evarestov, A A Berezin and G A Natanzon www.pdfgrip.com Chapter Introduction In the first chapter of this monograph we shall try, in so far as it is possible at the beginning of a book, to show how one can naturally and advantageously apply the theory of groups to the solution of physical problems We hope that this will help the reader who is mainly interested in the applications of group theory to physics to become familiar with the general ideas of abstract groups which are necessary for applications 1.1 Symmetry properties of physical systems It is frequently possible to establish the properties of physical systems in the form of symmetry laws These laws are expressed by the invariance (invariant form) of the equations of motion under certain definite transformations If, for example, the equations of motion are invariant under orthogonal transformations of Cartesian coordinates in three-dimensional space, it may be concluded that reference frames oriented in a definite way relative to each other are equivalent for the description of the motion of the physical system under consideration Equivalent reference frames are usually defined as frames in which identical phenomena occur in the same way when identical initial conditions are set up for them Conversely, if in a physical theory it is postulated that certain reference frames are equivalent, then the equations of motion should be invariant under the www.pdfgrip.com transformations relating the coordinates in these systems For example, the postulate of the theory of relativity which demands the equivalence of all reference frames moving with uniform velocity relative to one another is expressed by the invariance of the equations of motion under the Lorentz transformation The class of equivalent reference frames for a given problem is frequently determined from simple geometrical considerations applied to a model of the physical system This is done, for example, in the case of symmetric molecules, crystals and so on However, not all transformations under which the equations of motion are invariant can be interpreted as transformations to a new reference frame The symmetry of a physical system may not have an immediate geometrical interpretation For example, V A Fock has shown that the Schroedinger equation for the hydrogen atom is invariant under rotations in a four-dimensional space connected with the momentum space The symmetry properties of a physical system are general and very important features Their generality usually ensures that they remain valid while our knowledge of a given physical system grows They must not, however, be regarded as absolute properties; like any other descriptions of physical systems they are essentially approximate The approximate nature of some symmetry properties is connected with the current state of our knowledge, while in other cases it is due to the use of simplified models of physical systems which facilitate the solution of practical problems Thus, by the symmetry of a system we shall not always understand the invariance of its equations of motion under a certain set of transformations The following important property must always be remembered: if an equation is 10 www.pdfgrip.com COUNTEREXAMPLES IN ANALYSIS, Bernard R Gelbaum and John M H Olmsted These counterexamples deal mostly with the part of analysis known as “real variables.” The first half covers the real number system, and the second half encompasses higher dimensions 1962 edition xxiv+198pp ì 8ẵ 0-486-42875-3 CATASTROPHE THEORY FOR SCIENTISTS AND ENGINEERS, Robert Gilmore Advanced-level treatment describes mathematics of theory grounded in the work of Poincaré, R Thom, other mathematicians Also important applications to problems in mathematics, physics, chemistry and engineering 1981 edition References 28 tables 397 black-and-white illustrations xvii + 666pp ì 9ẳ 0-486-67539-4 COMPLEX VARIABLES: Second Edition, Robert B Ash and W P Novinger Suitable for advanced undergraduates and graduate students, this newly revised treatment covers Cauchy theorem and its applications, analytic functions, and the prime number theorem Numerous problems and solutions 2004 edition 224pp 6ẵ ì 9ẳ 0-486-46250-1 464 www.pdfgrip.com NUMERICAL METHODS FOR SCIENTISTS AND ENGINEERS, Richard Hamming Classic text stresses frequency approach in coverage of algorithms, polynomial approximation, Fourier approximation, exponential approximation, other topics Revised and enlarged 2nd edition 721pp ì 8ẵ 0-486-65241-6 INTRODUCTION TO NUMERICAL ANALYSIS (2nd Edition), F B Hildebrand Classic, fundamental treatment covers computation, approximation, interpolation, numerical differentiation and integration, other topics 150 new problems 669pp ì 8ẵ 0-486-65363-3 MARKOV PROCESSES AND POTENTIAL THEORY, Robert M Blumental and Ronald K Getoor This graduate-level text explores the relationship between Markov processes and potential theory in terms of excessive functions, multiplicative functionals and subprocesses, additive functionals and their potentials, and dual processes 1968 edition 320pp × 8½ 0-486-46263-3 ABSTRACT SETS AND FINITE ORDINALS: An Introduction to the Study of Set Theory, G B Keene This text unites logical and philosophical aspects of set theory in a 465 www.pdfgrip.com manner intelligible to mathematicians without training in formal logic and to logicians without a mathematical background 1961 edition 112pp ì 8ẵ 0-486-46249-8 INTRODUCTORY REAL ANALYSIS, A.N Kolmogorov, S V Fomin Translated by Richard A Silverman Self-contained, evenly paced introduction to real and functional analysis Some 350 problems 403pp ì 8ẵ 0-486-61226-0 APPLIED ANALYSIS, Cornelius Lanczos Classic work on analysis and design of finite processes for approximating solution of analytical problems Algebraic equations, matrices, harmonic analysis, quadrature methods, much more 559pp ì 8ẵ 0-486-65656-X AN INTRODUCTION TO ALGEBRAIC STRUCTURES, Joseph Landin Superb self-contained text covers “abstract algebra”: sets and numbers, theory of groups, theory of rings, much more Numerous well-chosen examples, exercises 247pp ì 8ẵ 0-486-65940-2 466 www.pdfgrip.com QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, V V Nemytskii and V.V Stepanov Classic graduate-level text by two prominent Soviet mathematicians covers classical differential equations as well as topological dynamics and ergodic theory Bibliographies 523pp ì 8ẵ 0-486-65954-2 THEORY OF MATRICES, Sam Perlis Outstanding text covering rank, nonsingularity and inverses in connection with the development of canonical matrices under the relation of equivalence, and without the intervention of determinants Includes exercises 237pp ì 8ẵ 0-486-66810-X INTRODUCTION TO ANALYSIS, Maxwell Rosenlicht Unusually clear, accessible coverage of set theory, real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, more Wide range of problems Undergraduate level Bibliography 254pp ì 8ẵ 0-486-65038-3 MODERN NONLINEAR EQUATIONS, Thomas L Saaty Emphasizes practical solution of problems; covers seven types of equations “ a welcome contribution to the existing literature . Math Reviews 490pp ì 8ẵ 467 www.pdfgrip.com 0-486-64232-1 MATRICES AND LINEAR ALGEBRA, Hans Schneider and George Phillip Barker Basic textbook covers theory of matrices and its applications to systems of linear equations and related topics such as determinants, eigenvalues and differential equations Numerous exercises 432pp × 8½ 0-486-66014-1 LINEAR ALGEBRA, Georgi E Shilov Determinants, linear spaces, matrix algebras, similar topics For advanced undergraduates, graduates Silverman translation 387pp ì 8ẵ 0-486-63518-X MATHEMATICAL METHODS OF GAME AND ECONOMIC THEORY: Revised Edition, Jean-Pierre Aubin This text begins with optimization theory and convex analysis, followed by topics in game theory and mathematical economics, and concluding with an introduction to nonlinear analysis and control theory 1982 edition 656pp × 9¼ 0-486-46265-X SET THEORY AND LOGIC, Robert R Stoll Lucid introduction to unified theory of mathematical concepts Set 468 www.pdfgrip.com theory and logic seen as tools for conceptual understanding of ì 8ẳ real number system 496pp 0-486-63829-4 Physics OPTICAL RESONANCE AND TWO-LEVEL ATOMS, L Allen and J H Eberly Clear, comprehensive introduction to basic principles behind all quantum optical resonance phenomena 53 illustrations Preface Index 256pp × 8½ 0-486-65533-4 QUANTUM THEORY, David Bohm This advanced undergraduate-level text presents the quantum theory in terms of qualitative and imaginative concepts, followed by specific applications worked out in mathematical detail Preface Index 655pp ì 8ẵ 0-486-65969-0 ATOMIC PHYSICS (8th EDITION), Max Born Nobel laureate’s lucid treatment of kinetic theory of gases, elementary particles, nuclear atom, wave-corpuscles, atomic structure and spectral lines, much more Over 40 appendices, bibliography 495pp × 8½ 0-486-65984-4 469 www.pdfgrip.com A SOPHISTICATE’S PRIMER OF RELATIVITY, P W Bridgman Geared toward readers already acquainted with special relativity, this book transcends the view of theory as a working tool to answer natural questions: What is a frame of reference? What is a “law of nature”? What is the role of the “observer”? Extensive treatment, written in terms accessible to those without a scientific background 1983 ed xlviii + 172pp ì 8ẵ 0-486-42549-5 AN INTRODUCTION TO HAMILTONIAN OPTICS, H A Buchdahl Detailed account of the Hamiltonian treatment of aberration theory in geometrical optics Many classes of optical systems defined in terms of the symmetries they possess Problems with detailed solutions 1970 edition xv + 360pp × 8½ 0-486-67597-1 PRIMER OF QUANTUM MECHANICS, Marvin Chester Introductory text examines the classical quantum bead on a track: its state and representations; operator eigenvalues; harmonic oscillator and bound bead in a symmetric force field; and bead in a spherical shell Other topics include spin, matrices, and the structure of quantum mechanics; the simplest atom; indistinguishable particles; and stationary-state perturbation theory 1992 ed xiv+314pp ì 9ẳ 0-486-42878-8 470 www.pdfgrip.com LECTURES ON QUANTUM MECHANICS, Paul A M Dirac Four concise, brilliant lectures on mathematical methods in quantum mechanics from Nobel Prize-winning quantum pioneer build on idea of visualizing quantum theory through the use of classical mechanics 96pp ì 8ẵ 0-486-41713-1 THIRTY YEARS THAT SHOOK PHYSICS: THE STORY OF QUANTUM THEORY, George Gamow Lucid, accessible introduction to influential theory of energy and matter Careful explanations of Dirac’s anti-particles, Bohr’s model of the atom, much more 12 plates Numerous drawings 240pp × 8½ 0-486-24895-X ELECTRONIC STRUCTURE AND THE PROPERTIES OF SOLIDS: THE PHYSICS OF THE CHEMICAL BOND, Walter A Harrison Innovative text offers basic understanding of the electronic structure of covalent and ionic solids, simple metals, transition metals and their compounds Problems 1980 edition 582pp ì 9ẳ 0-486-66021-4 HYDRODYNAMIC AND HYDROMAGNETIC STABILITY, S Chandrasekhar Lucid examination of the Rayleigh-Benard problem; clear coverage of the theory of instabilities causing convection 704pp ì 8ẳ 471 www.pdfgrip.com 0-486-64071-X INVESTIGATIONS ON THE THEORY OF THE BROWNIAN MOVEMENT, Albert Einstein Five papers (1905 — 8) investigating dynamics of Brownian motion and evolving elementary theory Notes by R Fỹrth 122pp ì 8ẵ 0-486-60304-0 THE PHYSICS OF WAVES, William C Elmore and Mark A Heald Unique overview of classical wave theory Acoustics, optics, electromagnetic radiation, more Ideal as classroom text or for self-study Problems 477pp ì 8ẵ 0-486-64926-1 GRAVITY, George Gamow Distinguished physicist and teacher takes reader-friendly look at three scientists whose work unlocked many of the mysteries behind the laws of physics: Galileo, Newton, and Einstein Most of the book focuses on Newton’s ideas, with a concluding chapter on post-Einsteinian speculations concerning the relationship between gravity and other physical phenomena 160pp × 8½ 0-486-42563-0 472 www.pdfgrip.com PHYSICAL PRINCIPLES OF THE QUANTUM THEORY, Werner Heisenberg Nobel Laureate discusses quantum theory, uncertainty, wave mechanics, work of Dirac, Schroedinger, Compton, Wilson, Einstein, etc 184pp × 8½ 0-486-60113-7 ATOMIC SPECTRA AND ATOMIC STRUCTURE, Gerhard Herzberg One of best introductions; especially for specialist in other fields Treatment is physical rather than mathematical 80 illustrations 257pp × 8½ 0-486-60115-3 AN INTRODUCTION TO STATISTICAL THERMODYNAMICS, Terrell L Hill Excellent basic text offers wide-ranging coverage of quantum statistical mechanics, systems of interacting molecules, quantum statistics, more 523pp × 8½ 0-486-65242-4 THEORETICAL PHYSICS, Georg Joos, with Ira M Freeman Classic overview covers essential math, mechanics, electromagnetic theory, thermodynamics, quantum mechanics, nuclear physics, other topics First paperback ì 8ẵ edition xxiii + 885pp 0-486-65227-0 473 www.pdfgrip.com PROBLEMS AND SOLUTIONS IN QUANTUM CHEMISTRY AND PHYSICS, Charles S Johnson, Jr and Lee G Pedersen Unusually varied problems, detailed solutions in coverage of quantum mechanics, wave mechanics, angular momentum, molecular spectroscopy, more 280 problems plus 139 supplementary exercises 430pp 6ẵ ì 9¼ 0-486-65236-X THEORETICAL SOLID STATE PHYSICS, Vol 1: Perfect Lattices in Equilibrium; Vol II: Non-Equilibrium and Disorder, William Jones and Norman H March Monumental reference work covers fundamental theory of equilibrium properties of perfect crystalline solids, non-equilibrium properties, defects and disordered systems Appendices Problems Preface Diagrams Index Bibliography Total of 1,301pp ì 8ẵ Two volumes Vol I: 0-486-65015-4 Vol II: 0-486-65016-2 WHAT IS RELATIVITY? L D Landau and G B Rumer Written by a Nobel Prize physicist and his distinguished colleague, this compelling book explains the special theory of relativity to readers with no scientific background, using such familiar objects as trains, rulers, and clocks 1960 ed ì 8ẵ vi+72pp 0-486-42806-0 474 www.pdfgrip.com A TREATISE ON ELECTRICITY AND MAGNETISM, James Clerk Maxwell Important foundation work of modern physics Brings to final form Maxwell’s theory of electromagnetism and rigorously derives his general equations of field theory 1,084pp × 8½ Two-vol set Vol I: 0-486-60636-8 Vol II: 0-486-60637-6 MATHEMATICS FOR PHYSICISTS, Philippe Dennery and Andre Krzywicki Superb text provides math needed to understand today’s more advanced topics in physics and engineering Theory of functions of a complex variable, linear vector spaces, much more Problems 1967 edition 400pp 6ẵ ì 9¼ 0-486-69193-4 INTRODUCTION TO QUANTUM MECHANICS WITH APPLICATIONS TO CHEMISTRY, Linus Pauling & E Bright Wilson, Jr Classic undergraduate text by Nobel Prize winner applies quantum mechanics to chemical and physical problems Numerous tables and figures enhance the text Chapter bibliographies Appendices Index 468pp ì 8ẵ 0-486-64871-0 METHODS OF THERMODYNAMICS, Howard Reiss Outstanding text focuses on physical technique of thermodynamics, typical problem areas of understanding, and 475 www.pdfgrip.com significance and use of thermodynamic potential 1965 ì 8ẵ edition 238pp 0-486-69445-3 THE ELECTROMAGNETIC FIELD, Albert Shadowitz Comprehensive under- graduate text covers basics of electric and magnetic fields, builds up to electromagnetic theory Also related topics, including relativity Over 900 problems 768pp ì 8ẳ 0-486-65660-8 GREAT EXPERIMENTS IN PHYSICS: FIRSTHAND ACCOUNTS FROM GALILEO TO EINSTEIN, Morris H Shamos (ed.) 25 crucial discoveries: Newton’s laws of motion, Chadwick’s study of the neutron, Hertz on electromagnetic waves, more Original accounts clearly annotated 370pp ì 8ẵ 0-486-25346-5 EINSTEINS LEGACY, Julian Schwinger A Nobel Laureate relates fascinating story of Einstein and development of relativity theory in well-illustrated, nontechnical volume Subjects include meaning of time, paradoxes of space travel, gravity and its effect on light, non-Euclidean geometry and curving of space-time, impact of radio astronomy and space-age discoveries, and more 189 b/w illustrations × 9¼ xiv+250pp 476 www.pdfgrip.com 0-486-41974-6 THE VARIATIONAL PRINCIPLES OF MECHANICS, Cornelius Lanczos Philosophic, less formalistic approach to analytical mechanics offers model of clear, scholarly exposition at graduate level with coverage of basics, calculus of variations, principle of virtual work, equations of motion, more 418pp ì 8ẵ 0-486-65067-7 Paperbound unless otherwise indicated Available at your book dealer, online at www.doverpublications.com, or by writing to Dept GI, Dover Publications, Inc., 31 East 2nd Street, Mineola, NY 1501 For current price information or for free catalogues (please indicate field of interest), write to Dover Publications or log on to www.doverpublications.com and see every Dover book in print Dover publishes more than 400 books each year on science, elementary and advanced mathematics, biology, music, art, literary history, social sciences, and other areas 477 www.pdfgrip.com Editor’s note The group O + (n) falls into two separate parts which cannot be transformed continuously into each other: these are the sets of real orthogonal matrices with determinant + and −1 respectively For such groups the relations (10.1) and (10.2) require generalization, as many of the statements of this chapter 478 www.pdfgrip.com ... www.pdfgrip.com The set of all non-singular n-th order matrices (or the corresponding linear transformations in n-dimensional space), GL(n), is an example of a non-Abelian group It is clear that the elements... number of elements in a group is finite, then the group is called a finite group; if the number of elements is infinite, the group is called an infinite group The number of elements in a finite... rotation group we obtain the orthogonal group O (3) Molecular symmetry groups, i.e point groups, consist of certain orthogonal transformations of three-dimensional space For example, the symmetry group